generators of boosts. We have
so that different “boosts” commute with each other. The Galilei transformations thus correspond to an abelian Lie group. In particular
Boosts
Let S and S′ be two inertial frames whose clocks are synchronized in such a way, that their respective origins coincide at time t = 0. Then we have for an eigenstate of the momentum operator, as seen by observers O and O′ in S and S′
respectively. In particular, for a particle at rest in system S we obtain, from the point of view of O′,
Define
where
We have the following property of Galilei transformations not shared by Lorentz transformations (compare with (2.23)): boosts and rotations separately form a group. Indeed one easily checks that
(4)Causality
We next want to show that NRQM violates the principle of causality. We have for any interacting theory,
Let |En
Then
Define
as well as
The kernel
In terms of this kernel we have from above,
We now specialize to the case of a free point-like particle. In that case
and correspondingly we have with (1.8),
Notice that the kernel K0 satisfies the desired initial condition (1.9).
From (1.10), for the initial condition
or in particular
that is, for an infinitesimal time after t0 one already finds the particle with equal probability anywhere in space; this violates obviously the principle of relativity, as well as causality.
________________________
1In a relativistic theory,
2In Quantum Mechanics
In this chapter we use everywhere lower indices, repeated indices being summed over.
3For notational simplicity we suppose the spectrum to be discrete.
Chapter 2
Lorentz Group and Hilbert Space
In this chapter we first discuss the realization of the homogeneous Lorentz transformations in four-dimensional space-time, as well as the corresponding Lie algebra. From here we obtain all finite dimensional representations, and in particular the explicit form of the matrices representing the boosts for the case of spin = 1/2, which will play a fundamental role in Chapter 3. The Lorentz transformation properties of massive and zero-mass 1-particle in Hilbert space (and their explicit realization in Chapter 9) lie at the heart of the Fock space representation (second quantization) in Chapter 9. It is assumed that the reader is already familiar with the essentials of the Special Theory of Relativity and of Group Theory.
2.1Defining properties of Lorentz transformations
Homogeneous Lorentz transformations are linear transformations on the space-time coordinates,1
leaving
